Solutions of Eq

Lane improved on these tables with accurate polynomial fits to numerical solutions of Eq. 11-17 [16]. Two equations result the first is applicable when rja 2... 

Equations (16)-(20) show that the real adiabatic eigenstates are everywhere smooth and continuously differentiable functions of Q, except at degenerate points, such that E (Q) — E, [Q) = 0, where, of com se, the x ) are undefined. There is, however, no requirement that the x ) should be teal, even for a real Hamiltonian, because the solutions of Eq. fl4) contain an arbitrary Q dependent phase term, gay. Second, as we shall now see, the choice that x ) is real raises a different type of problem. Consider, for example, the model Hamiltonian in Eq. (8), with / = 0 ... 

The presence of this term can also introduce numerical inefficiency problems in the solution of Eq. (31). Since the ADT matrix U(qx ) is arbifiary, it can be chosen to make Eq. (31) have desirable properties that Eq. (15) does not possess. The parameter U(qx) can, for example, be chosen so as to automatically minimize W (Rx) relative to everywhere in internal... 

The —(/i /2p)W (Rx) matrix does not have poles at conical intersection geometries [as opposed to W (R )] and furthermore it only appears as an additive term to the diabatic energy matrix (q ) and does not increase the computational effort for the solution of Eq. (55). Since the neglected gradient term is expected to be small, it can be reintroduced as a first-order perturbation afterward, if desired. 

The diabatic LHSFs are not allowed to diverge anywhere on the half-sphere of fixed radius p. This boundary condition furnishes the quantum numhers n - and each of which is 2D since the reference Hamiltonian hj has two angular degrees of freedom. The superscripts n(, Q in Eq. (95), with n refering to the union of and indicate that the number of linearly independent solutions of Eqs. (94) is equal to the number of diabatic LHSFs used in the expansions of Eq. (95). 

If V(R) is known and the mahix elements ffap ate evaluated, then solution of Eq. nO) for a given initial wavepacket is the numerically exact solution to the Schrddinger equation. 

In this section, diabatization is formed employing the adiabatic-to-diabatic transformation matrix A, which is a solution of Eq. (19). Once A is calculated, the diabatic potential matiix W is obtained from Eq. (22). Thus Eqs. (19) and (22) form the basis for the procedure to obtain the diabatic potential matrix elements. 

With the modified expression we can now extend the solution of Eq. (162) to any number of conical intersections. The solution in Eq. (162) stands for a single conical intersection located at an arbitrary point ( yo,0 /o)- Since ta(, 0) and T,(<7,0) are scalars the solution in case of N conical intersections located at the... 

S e is the family of solutions of Eq. (1) with initial states due to Eq. (2) and Eq. (21). The initial quantum state tjj, is assumed to be independent from e with only finitely mm y energy levels Ek, k = 1,..., n being initially excited. 

This equation is a quadratic and has two roots. For quantum mechanical reasons, we are interested only in the lower root. By inspection, x = 0 leads to a large number on the left of Eq. (1-10). Letting x = leads to a smaller number on the left of Eq. (1-10), but it is still greater than zero. Evidently, increasing a approaches a solution of Eq. (1-10), that is, a value of a for which both sides are equal. By systematically increasing a beyond 1, we will approach one of the roots of the secular matrix. Negative values of x cause the left side of Eq. (1-10) to increase without limit hence the root we are approaching must be the lower root. 

By Cramer s rule, each solution of Eqs. (2-44) is given as the ratio of determinants... 

These are general equations that do not depend on the particular mixing rules adopted for the composition dependence of a and b. The mixing rules given by Eqs. (4-221) and (4-222) can certainly be employed with these equations. However, for purposes of vapor/liquid equilibrium calculations, a special pair of mixing rules is far more appropriate, and will be introduced when these calculations are treated. Solution of Eq. (4-232) for fugacity coefficient at given T and P reqmres prior solution of Eq. (4-231) for V, from which is found Z = PV/RT. 

When q is zero, Eq. (5-18) reduces to the famihar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplaces equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations apphcable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the Introduction. The methods may also be extended to three-dimensional problems. 

The definition of the heat-transfer coefficient is arbitrary, depending on whether bulk-fluid temperature, centerline temperature, or some other reference temperature is used for ti or t-. Equation (5-24) is an expression of Newtons law of cooling and incorporates all the complexities involved in the solution of Eq. (5-23). The temperature gradients in both the fluid and the adjacent solid at the fluid-solid interface may also be related to the heat-transfer coefficient ... 

Solution of Eq. (6-114) for G and differentiation with respect to p reveals a maximum mass flux = P2VMJ RT) and a corresponding exit velocity and exit Mach number Mo = L/. This... 

The calculation of (DCFRR) usually requires a trial-and-error solution of Eq. (9-57), hut rapidly convergent methods are avadahle [N. H. Wild, Chem. Eng, 83, 15.3-154 (Apr. 12, 1976)]. For simplicity linear interpolation is often used. 

FIG. 14-5 Nnmher of overall gas-phase mass-transfer units in a packed absorption tower for constant mGf /LM solution of Eq. (14-23). (From Sherwood and Pigford, Absorption and Extraction, McGraw-Hill, New York, 1952. )... 

Isocratic Elution In the simplest case, feed with concentration cf is apphed to the column for a time tp followed by the pure carrier fluid. Under trace conditions, for a hnear isotherm with external mass-transfer control, the linear driving force approximation or reaction kinetics (see Table 16-12), solution of Eq. (16-146) gives the following expression for the dimensionless solute concentration at the column outlet ... 

The time-dependent nature of the emergency pressure relieving event is obtained by the simultaneous solution of Eqs. (26-27) and (26-28). Generally, the only unknown parameters in these two equations are the venting rate W and the vent stream quahty (Xo). The vent rate W at any instant is a func tion of the upstream conditions and the relief system geometry. 

A number of papers have explored methods for the solution of Eqs. (26-29) and (26-31), especially for the two-phase conditions. The reader is referred to the DIERS Project Manual for a more detailed review and list of appropriate references and available computer programs. 

The second physical quantity of interest is, r t = 90 pm, the critical crack tip stress field dimension. Irwin s analysis of the crack tip process zone dimension for an elastic-perfectly plastic material began with the perfectly elastic crack tip stress field solution of Eq. 1 and allowed for stress redistribution to account for the fact that the near crack tip field would be limited to Oj . The net result of this analysis is that the crack tip inelastic zone was nearly twice that predicted by Eq. 3, such that... 

In steady-state conditions the right side of Eq. (4.180) is zero, and no heat generation takes place the thermal conductivity in the one-dimensional case is constant. The solution of Eq. (4.182) is... 

Joint solution of Eqs. (7.170) and (7.178) allows one to calculate the maximum amount of heat supplied by a directing jet with the assumption that the jet reaches the occupied zone > 0-1 m/s) and (fgp -fgP)/Tgp is less than 0.2 at the point where it enters the occupied zone. The maximum initial temperature difference of the air supplied by vertical directing jet is... 

The initial speed of the air supplied by the air curtain is determined according to the following universal dependence, which results from the joint solution of Eqs. (7.213)-(7.216) ... 

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